I have a probability density function over $SO\left(3\right)$, which I am trying to sample from. The $pdf$ is given as a generalized fourier series:
$$ f\left(\omega,\theta,\phi\right)=\sum s_{\lambda}^{n}Z_{\lambda}^{n}\left(\omega,\theta,\phi\right)$$
where $s_{\lambda}^{n}$ are the coefficients and $Z_{\lambda}^{n}\left(\omega,\theta,\phi\right)$ are basis functions (symmetrized hyperspherical harmonics), and $SO\left(3\right)$ is parameterized by the variables $\left(\omega,\theta,\phi\right)$, which are the rotation angle, and spherical coordinates of the rotation axis, respectively. I want to sample values of $\left(\omega,\theta,\phi\right)$ (i.e. rotations) from this distribution, but I'm having trouble doing so.
So far I have essentially tried two methods: rejection sampling, a discrete method.
Both methods have given me something that is qualitatively similar to what I would expect, but they seem to have an erroneous uniform distribution superimposed. So I have two questions:
(1) Any ideas why I might be getting this uniform noise?
(2) Any suggestions of how to fix it or a better way to sample from this kind of distribution?
Further Details: To check my sampling method I used some software to generate a known distribution and sample rotations. I then computed $s_{\lambda}^{n}$ from these "correct samples". Then I tried to generate samples myself from the spectral form of $f\left(\omega,\theta,\phi\right)$ given above. Next I calculated $s_{\lambda}^{n}$ from my samples and compared the two. My samples led to the low order terms being generally too small in magnitude and the higher order terms being too large in magnitude.
The discrete method that I used consisted of generating a "grid" of points over $SO\left(3\right)$, and using these as the centers of bins. The bins were sized proportional to the probability density at the bin center. Then uniform samples were generated and the number that fell in each bin was then proportional to the associated probability of the respective bins.
Again, both methods produced a sort of background noise which looks like a uniform distribution on top of the correct distribution.
As, a side note, it seems like I ought to be able to exploit the form of this expression for efficient sampling, but I haven't made use of the spectral decomposition at all in my attempts.